Question

Form the pair of linear equations for the following problems and find their solution by substitution method.

(i) The difference between two numbers is 26 and one number is three times the other. Find them.

(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km.

(v) A fraction becomes\(\frac{ 9}{11}\), if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes \(\frac{5}{6}\). Find the fraction.

(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

Answer 1

(i) Let the two numbers be x and y, with y>x. The problem states y = 3x (1) and y - x = 26 (2). Substituting (1) into (2) yields 3x - x = 26, which simplifies to 2x = 26, so x = 13. Substituting x = 13 into (1) gives y = 3 * 13 = 39. Therefore, the numbers are 13 and 39.

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(ii) Let the larger angle be x and the smaller angle be y. Supplementary angles sum to 180°. The problem gives x + y = 180° (1) and x - y = 18° (2). From (1), x = 180° - y (3). Substituting (3) into (2) results in (180° - y) - y = 18°, simplifying to 180° - 2y = 18°. Rearranging gives 2y = 180° - 18° = 162°, so y = 81°. Substituting y = 81° into (3) gives x = 180° - 81° = 99°. Thus, the angles are 99° and 81°.

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(iii) Let the cost of a bat be x and the cost of a ball be y. The given information translates to 7x + 6y = 3800 (1) and 3x + 5y = 1750 (2). From (1), y = (3800 - 7x) / 6 (3). Substituting (3) into (2) gives 3x + 5((3800 - 7x) / 6) = 1750. Multiplying by 6 to clear the fraction: 18x + 5(3800 - 7x) = 10500. This simplifies to 18x + 19000 - 35x = 10500. Combining terms, -17x = 10500 - 19000, so -17x = -8500, which means x = 500. Substituting x = 500 into (3) yields y = (3800 - 7 * 500) / 6 = (3800 - 3500) / 6 = 300 / 6 = 50. Therefore, the cost of a bat is Rs 500 and the cost of a ball is Rs 50.

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(iv) Let the fixed charge be Rs x and the per km charge be Rs y. The problem states x + 10y = 105 (1) and x + 15y = 155 (2). From (1), x = 105 - 10y (3). Substituting (3) into (2) gives (105 - 10y) + 15y = 155. Simplifying, 105 + 5y = 155, so 5y = 155 - 105 = 50, which means y = 10. Substituting y = 10 into (3) gives x = 105 - 10 * 10 = 105 - 100 = 5. Hence, the fixed charge is Rs 5 and the per km charge is Rs 10. The charge for 25 km is x + 25y = 5 + 25 * 10 = 5 + 250 = Rs 255.

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(v) Let the fraction be x/y. The problem states (x+2)/(y+2) = 9/11 and (x+3)/(y+3) = 5/6. Cross-multiplying the first equation gives 11(x+2) = 9(y+2), which simplifies to 11x + 22 = 9y + 18, or 11x - 9y = -4 (1). Cross-multiplying the second equation gives 6(x+3) = 5(y+3), which simplifies to 6x + 18 = 5y + 15, or 6x - 5y = -3 (2). From (1), x = (-4 + 9y) / 11 (3). Substituting (3) into (2) yields 6((-4 + 9y) / 11) - 5y = -3. Multiplying by 11 to clear the fraction: 6(-4 + 9y) - 55y = -33. This simplifies to -24 + 54y - 55y = -33, so -y = -33 + 24, which means -y = -9, so y = 9. Substituting y = 9 into (3) gives x = (-4 + 9 * 9) / 11 = (-4 + 81) / 11 = 77 / 11 = 7. Therefore, the fraction is 7/9.

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(vi) Let Jacob's current age be x and his son's current age be y. Five years from now, their ages will be x+5 and y+5. The problem states (x+5) = 3(y+5), which simplifies to x + 5 = 3y + 15, or x - 3y = 10 (1). Five years ago, their ages were x-5 and y-5. The problem states (x-5) = 7(y-5), which simplifies to x - 5 = 7y - 35, or x - 7y = -30 (2). From (1), x = 3y + 10 (3). Substituting (3) into (2) gives (3y + 10) - 7y = -30. Simplifying, -4y + 10 = -30, so -4y = -40, which means y = 10. Substituting y = 10 into (3) gives x = 3 * 10 + 10 = 30 + 10 = 40. Therefore, Jacob's current age is 40 years, and his son's current age is 10 years.